I don’t think that the question-begging charge sticks on Lewis (or rather, on me), because I don’t think that Williams gets to phrase his response in the way in which Bill puts it [in the original handout].  Williams doesn’t have any reason to bracket the ignored possibilities with the sotto voce proviso, because he doesn’t accept the Rule of Attention in the first place.  Williams does say that the Rule is “anything but trivial,” and Lewis wouldn’t disagree.  What Lewis does say is that the Rule follows as a triviality from the way the sotto voce proviso is phrased. (p. 232)  And he claims that the fallibilist will not accept this, and will say something very much like what Williams does say; “And then you will insist that those far-fetched possibilities of error that we attend to at the behest of the sceptic are nonetheless possibilities we could have properly ignored.  You will say that no amount of attention can, by itself, turn them into relevant alternatives.”  So the question is, does one have to be a ‘contented fallibilist’ to say this, or can someone say this and be an infallibilist too?
     I don’t think so.  Suppose we have a candidate for knowledge of P, and a possibility in which not-P.  The possibility is one we are all aware of (including the candidate).  The possibility is not eliminated by the subject’s evidence.  For Lewis the question is answered: the candidate does not know P.  For Williams there is a further question about whether this is a possibility that we couldn’t have properly ignored.  Suppose we decide we could have properly ignored it; what do we do now?  Cease to pay attention to it; ignore it, in the proper sense of the word which, as Williams notes, implies prior recognition.  So it is still around (as a matter of psychological fact), but we have decided to pay it no notice (normative decision).  So we are entitled to say something like, “S knows that P, even though there is a possibility in which not-P uneliminated by S’s evidence, because it isn’t worth paying attention to; we could have properly ignored it.”  But that looks like a fallibilist knowledge-claim to me.  Williams would have to insist on continuing to bracket the not-P possibility with the sotto voce proviso to remain infallibilist; but isn’t that just an implicit acceptance of the Rule of Attention?
     (Maybe Williams wouldn’t think it would be, because he seems to have the point of the proviso for Lewis not quite right.  On p. 12 he says: “I take it that the proviso is sotto voce because an explicit relativization to context is not (intuitively) part of the content of an everyday knowledge-claim.  This seems right.” But it involves more than that: everyday knowledge-claims can’t be explicitly relativized to everyday contexts, whether they normally are or not, because doing so would call attention (in the psychological sense) to other contexts, and thus other error possibilities.  And then we’d either have to eliminate them through the evidence, or else be willing to say that they could have been properly ignored, and thus be willing to make knowledge-claims which explicitly deny the relevance of those possibilities, which again is a fallibilist thing to do.)
     Even though Lewis says that the sotto voce proviso could have been stated differently, I’m not sure that doing so wouldn’t remove the need for it to be sotto voce.  After all, if we’re talking about what could have been properly ignored, without the Rule of Attention, it doesn’t matter whether our attention is called to those possibilities.  The fact that the proviso is sotto voce depends on the ability of the possibilities’ being called to mind to immediately become relevant and unignorable.  But if the possibilities aren’t bracketed sotto voce, if they’re part of the knowledge-condition itself, then the knowledge-condition is fallibilist.  And we’re back to the standoff.
     I suppose one might try to hang onto the idea of infallibility by rejecting closure.  But actually I don’t see how that helps.  Even if knowing that P and knowing that ‘P implies not-H’ don’t entail knowing that not-H, the fact that not-H is broached as a possibility means that one can say something like “I know that P even though I don’t know that not-H,” which is fallibilist.  As long as one thinks that noticed possibilities aren’t necessarily relevant, and might have been properly ignored, one ought to be able to include them as part of a knowledge-claim: “S knows P despite not being able to rule out H, because H might have been properly ignored.”  But that just looks like a fallibilist claim, and I’m not sure how one might go about denying that it is.
 Maybe this just shows how inoffensive fallibilism is.  After all, one might look at the above and think “if that’s fallibilism, then I don’t mind being a fallibilist, because after all I merely want to rule out the relevant alternatives, and if I can ignore some alternative then it isn’t relevant.”  Williams seems to think something like this when he suggests that our infallibilist intuitions are leftovers from the “discarded ideal of certainty.”  I’m not necessarily contesting that, but I do still think that his position is fallibilist, and so to the extent that one is attracted to infalliblism one has reason to accept the Rule of Attention.
    As a side note, Lewis might also quarrel with Williams’ implication that the Rule of Attention is a theory-driven addition for him (Williams p. 16).  Williams suggests that that’s the only way to get skepticism into the picture at all, and seems to imply that that betrays something wrong with Lewis’ view.  But why should this be thought especially problematic for Lewis?  Skeptical hypotheses have no practical implications; if they did they wouldn’t be skeptical hypotheses, because we would be able in principle to tell whether they were true or not by attending to those implications.  So it isn’t as if the question of why skepticism ever seems relevant is somehow specific to Lewis’s account.  Anyone who addresses skeptical concerns has to have a way of saying why skeptical possibilities ought to be considered, and those can’t depend on any practical implications such possibilities might have, because by hypothesis they don’t have any.
    I don’t think any of the foregoing is going to be especially moving to Williams, because I don’t think that he has any pretensions to infallibilism.  (See his discussion on p. 9; “Now even assuming that we have infallibilist intuitions, it is not obvious that they should be taken seriously …. Lewis’ “scepticism” is just fallibilism as it ought to be understood.” Also later, on p. 16-7, where he ends section 6 by rejecting “Lewis’ attenuated conception of epistemology….”)  While he doesn’t come out in favor of fallibilism, he seems sympathetic to it, and seems to think that part of what is ‘attenuated’ about Lewis’ conception of epistemology is this insistence on infallibility.  But be that as it may, I think the Rule is essential for an infallibilist position, and so to the extent one finds anything right about infallibilism one has reason to accept the Rule, and so Lewis isn’t (or rather, I’m not) begging the question in citing his infallibilist argument in support of it.